Triangles and quadrilaterals are familiar shapes you've seen since elementary school, but in middle school you move on to logically explaining "why" they behave the way they do. There are properties that never change no matter how much you change the size or angles, and knowing them lets you solve geometry problems much faster.
Let's start with the isosceles triangle. It's a triangle where two sides are equal in length, and what's interesting is that from the single condition "two sides are equal in length," the conclusion "the two angles opposite those sides (the base angles) are equal too" follows automatically. A condition about side length ends up determining an entirely different property — angle size. To prove this, you drop a perpendicular from the apex to the base, folding the triangle exactly in half; the fact that the two halves match up perfectly when folded (congruence) reveals that the two base angles are equal.
Moving on to quadrilaterals, the parallelogram is the most basic shape. As long as both pairs of opposite sides are parallel, it's called a parallelogram. From this one condition alone, a whole string of properties follows: "opposite sides are equal in length," "opposite angles are equal in size," and "the two diagonals split each other exactly in half (bisect each other)." The charm of a parallelogram is that this many properties arise automatically just from the single fact of being parallel, with no extra special conditions needed.
Add one more condition on top of a parallelogram and you get an even more special quadrilateral. Make all four angles 90° and you get a rectangle, which also gains the additional property that "the two diagonals are equal in length." Conversely, make all four sides equal in length and you get a rhombus, which gains the property that "the two diagonals meet at a right angle." And satisfy both conditions at once (all four angles 90°, all four sides equal) and you get a square — a square is essentially both a rectangle and a rhombus at the same time, so it inherits every property of both shapes.
Finally, the trapezoid has a much looser condition than a parallelogram. As long as exactly one pair of sides is parallel, it can be called a trapezoid. So a trapezoid doesn't care whether its legs (the two slanted sides) are different lengths, or what the angles are. Where a parallelogram has the strict condition of "both pairs parallel," a trapezoid has the much looser condition of "just one pair parallel."
Looking at it this way, there's a containment relationship among quadrilaterals. A square belongs to both the rhombus family and the rectangle family, and both the rhombus and the rectangle belong to the parallelogram family. Much like how, in animal classification, "dogs are inside mammals, and Jindo dogs are inside dogs," the more conditions you add, the narrower and more special the shape becomes. When solving a geometry problem, figuring out exactly which category a given quadrilateral belongs to first lets you use every property that shape has all at once — which is a huge advantage.
On our activity page, you can freely reshape each shape with sliders and watch matching side lengths and matching angle sizes get marked in color. See for yourself that no matter how you change the shape, the marked properties never break down.